INVARIANT DIFFERENTIAL OPERATORS ON SP_{2} × ℝ^{(1,2)}

Jae-Hyun Yang

1. Introduction

We let

SL_{2,1}(ℝ) := SL(2, ℝ) ⋉ ℝ^{(1,2)}

be the semidirect product of the special linear group SL(2, ℝ) of degree 2 and the commutative group ℝ^{(1,2)} equipped with the following multiplication law

(1.1)  (g, α) ∗ (h, β) := (gh, α^{t}h^{−1} + β), g, h ∈ SL(2, ℝ), α, β ∈ ℝ^{(1,2)}.

Throughout this paper, for two positive integers m and n, we denote by F^{(m,n)} the set of all m × n matrices with entries in a commutative ring F. We let

SP_{2} := { Y ∈ ℝ^{(2,2)} | Y = ^{t}Y > 0, det Y = 1 }

be the symmetric space consisiting of all positive symmetric 2×2 real matrices with det Y = 1. Then it is easy to see that SL_{2,1}(ℝ) acts on SP_{2} × ℝ^{(1,2)} transitively by

(1.2)  ( g, α) · ( Y, V) := ( gY ^{t}g, ( V + α)^{t}g),

where g ∈ SL(2, ℝ), α ∈ ℝ^{(1,2)}, Y ∈ SP_{2} and V ∈ ℝ^{(1,2)}. Let

K := { ( k, 0 ) ∈ SL_{2,1}(ℝ) | k ∈ SO(2) }

be the subgroup of SL_{2,1}(ℝ), where SO(2) denotes the special orthogonal group of degree 2. Then it is easy to see that K is a maximal compact subgroup of SL_{2,1}(ℝ) stabilizing the origin (E_{2}, 0). Thus SP_{2} × ℝ^{(1,2)} may be identified with the homogeneous space SL_{2,1}(ℝ)/K as follows :

(1.3)  SL_{2,1}(ℝ)/K ∋ (g, α)K ⟼ (g, α) · (E_{2}, 0) ∈ SP_{2} × ℝ^{(1,2)},

where g ∈ SL(2, ℝ) and α ∈ ℝ^{(1,2)}.

The motivation that we study the action (1.2) of SL_{2,1}(ℝ) on the homogeneous space SP_{2} × ℝ^{(1,2)} is described in [Y7]. However for the convenience of the reader, we explain the motivation roughly.

Typeset by AMS-TEX

This work was supported financially by TGRC-KOSEF and Max-Planck Institut für Mathematik.

1991 Mathematics Subject Classification. Primary 11F55, 32C55, 32N10, 43A85, 43A90.

Let n and m be two fixed positive integers. Let

Sp(n, ℝ) = { M ∈ ℝ^{(2n,2n)} | ^{t}MJ_{n}M = J_{n} }

be the real symplectic group of degree n, where

J_{n} := ( \begin{matrix}0 & E_{n} \\ −E_{n} & 0 \end{matrix} ) .

Here we denote by ^{t}A the transpose of any matrix A. Let

H_{n} := { Z ∈ ℂ^{(n,n)} | Z = ^{t}Z, Im Z > 0 }

be the Siegel upper half plane of degree n. Then Sp(n, ℝ) acts on H_{n} transitively by

(1.4)  M < Z >:= (AZ + B)(CZ + D)^{−1},

wher M = ( \begin{matrix}A & B \\ C & D \end{matrix} ) ∈ Sp(n, ℝ) and Z ∈ H_{n}. We let

P_{n} := { Y ∈ ℝ^{(n,n)} | Y = ^{t}Y > 0 }

be the open convex cone of positive definite matrices of degree n in the Euclidean space ℝ^{n(n+1)/2}. Then the action (1.4) gives a natural action of the general linear group GL(n, ℝ) on P_{n} defined by

(1.5)  (g, Y) ⟼ g < Y >:= gY ^{t}g, g ∈ GL(n, ℝ), Y ∈ P_{n}.

We consider the Heisenberg group

H_{ℝ}^{(n,m)} := { (λ, μ; κ) | λ, μ ∈ ℝ^{(m,n)}, κ ∈ ℝ^{(m,m)}, κ + μ^{t}λ symmetric }

endowed with the following multiplication law

(λ, μ; κ) ∘ (λ′, μ′; κ′) := (λ + λ′, μ + μ′, κ + κ′ + λ^{t}μ′ − μ^{t}λ′).

We define the semidirect product of Sp(n, ℝ) and H_{ℝ}^{(n,m)}

(1.6)  G_{n,m}^{J}(ℝ) := Sp(n, ℝ) ⋉ H_{ℝ}^{(n,m)}

endowed with the following multiplication law

(M, (λ, μ; κ)) · (M′, (λ′, μ′; κ′))
:= (MM′, (λ̃ + λ′, μ̃ + μ′,; κ + κ′ + λ̃^{t}μ′ − μ̃^{t}λ′)),

where M, M′ ∈ Sp(n, ℝ) and (λ̃, μ̃) := (λ, μ)M′. The group G_{n,m}^{J}(ℝ) is called the Jacobi group of degree (n, m). It is easy to see that the Jacobi group G_{n,m}^{J}(ℝ) acts on the homogeneous space H_{n} × ℂ^{(m,n)} transitively by

(1.7)  (M, (λ, μ; κ)) · (Z, W) := (M < Z >, (W + λZ + μ)(CZ + D)^{−1}),

where M = ( \begin{matrix}A & B \\ C & D \end{matrix} ) ∈ Sp(n, ℝ), (λ, μ; κ) ∈ H_{ℝ}^{(n,m)} and (Z, W) ∈ H_{n} × ℂ^{(m,n)}.

The action (1.7) arises from the study of Jacobi forms ( cf. [E-Z], [Y1]-[Y6] ) and plays an important role in the study of the moduli space of principally polarized abelian varieties. We let

(1.8)  G_{n,m}(ℝ) := GL(n, ℝ) ⋉ ℝ^{(m,n)}

be the semidirect product of GL(n, ℝ) and the commutative additive group ℝ^{(m,n)} equipped with the following multiplication law

(1.9)  (g, a) ∗ (h, b) := (gh, a^{t}h^{−1} + b),

where g, h ∈ GL(n, ℝ) and a, b ∈ ℝ^{(m,n)}. Then the action (1.7) of G_{n,m}^{J}(ℝ) on H_{n} × ℂ^{(m,n)} gives a canonical action of G_{n,m}(ℝ) on the reductive homogeneous space P_{n} × ℝ^{(m,n)} given by

(1.10)  (g, a) · (Y, V) := (gY ^{t}g, (V + a)^{t}g),

where g ∈ GL(n, ℝ), a ∈ ℝ^{(m,n)}, Y ∈ P_{n} and V ∈ ℝ^{(m,n)}.

We let

SL_{n,m}(ℝ) := SL(n, ℝ) ⋉ ℝ^{(m,n)}

be the semidirect product of the special linear group SL(n, ℝ) and the commutative additive group ℝ^{(m,n)} with the multiplication law (1.9). Obviously SL_{n,m}(ℝ) is a subgroup of G_{n,m}(ℝ).

We let

SP_{n} := { Y ∈ P_{n} | det Y = 1 }

be the symmetric space. Clearly SL_{n,m}(ℝ) acts on SP_{n} × ℝ^{(m,n)} transitively by (1.10). Using this action, the author introduced a new notion of Maass -Jacobi forms ( [Y7-8] ) generalizing that of Maass wave forms ( cf. [G], [I], [H] and [T] ).

In this paper, we consider only the simplest case n = 2 and m = 1. The main purpose of this paper is to find all the differential operators on SP_{2}×ℝ^{(1,2)} invariant under the action (1.2) explicitly. This paper is organized as follows. In section 2, we prove the main theorem, that is, we give a complete description of the algebra of differential operators on SP_{2} × ℝ^{(1,2)} invariant under the action (1.2). And we provide a SL_{2,1}(ℝ)-invariant Riemannian metric on SP_{2} × ℝ^{(1,2)} and compute its Laplacian. In the final section, we calculate the derivatives for functions on SL_{2,1}(ℝ) and SP_{2} × ℝ^{(1,2)} explicitly.

Acknowledgement. I would like to give my deep thanks to Max-Planck Institut für Mathematik for its financial support and its wonderful hospitality during my stay in Bonn.

Notations. We denote by ℤ, ℝ and ℂ the ring of integers, the field of real numbers and the field of complex numbers respectively. ℤ^{+} denotes the set of all positive integers. The symbol “:=” means that the expression on the right is the definition of that on the left. F^{(k,l)} denotes the set of all k × l matrices with entries in a commutative ring F. For a square matrix A, σ(A) denotes the trace of A. For any M ∈ F^{(k,l)}, ^{t}M denotes the transpose of M. For A ∈ F^{(k,l)} and B ∈ F^{(k,k)}, we set B[A] = ^{t}ABA. E_{n} denotes the identity matrix of degree n. ℍ denotes the Poincaré upper-half plane.

2. Proof of Main Theorem

From now on, we write G := SL_{2,1}(ℝ) for brevity. We recall that SL(2, ℝ) acts on ℍ transitively by

(2.1)  g < τ >:= (aτ + b)(cτ + d)^{−1}, ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL(2, ℝ), τ ∈ ℍ.

Now we define the action of G on ℍ × ℂ by

(2.2)  (g, α) ∘ (τ, z) := (^{t}g^{−1} < τ >, (z + α_{1}τ + α_{2})(−bτ + a)^{−1} ),

where g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL(2, ℝ) and α = (α_{1}, α_{2}) ∈ ℝ^{(1,2)}, and (τ, z) ∈ ℍ×ℂ. Since

the action (2.2) is transitive and K is the stabilizer of this action at the origin (i, 0), ℍ × ℂ can be identified with the homogeneous space G/K as follows :

(2.3)  G/K ∋ (g, α)K ⟼ (g, α) ∘ (i, 0).

We observe that we can express an element Y of SP_{2} uniquely as

(2.4)  Y = ( \begin{matrix}y^{−1} & 0 \\ 0 & y \end{matrix} ) [( \begin{matrix}1 & −x \\ 0 & 1 \end{matrix} )] = ( \begin{matrix}y^{−1} & −xy^{−1} \\ −xy^{−1} & x^{2}y^{−1} + y \end{matrix} )

with x, y ∈ ℝ and y > 0.

Proposition 2.1. We define the mapping T : SP_{2} × ℝ^{(1,2)} ⟶ ℍ × ℂ by

(2.5)  T (Y, V) := (x + iy, v_{1}(x + iy) + v_{2} ),

where Y is of the form (2.4) and V = (v_{1}, v_{2}) ∈ ℝ^{(1,2)}. Then the mapping T is a bijection which is compatible with the above two actions (1.2) and (2.2).

Proof. It is easy to prove the above proposition and so we omit its proof. □

Now we give a complete description of the algebra 𝔻(SP_{2} × ℝ^{(1,2)}) of all differential operators on SP_{2} × ℝ^{(1,2)} invariant under the action (1.2) of G. First we note that the Lie algebra 𝔤 of G is given by

𝔤 = { (X, Z) | X ∈ ℝ^{(2,2)}, σ(X) = 0, Z ∈ ℝ^{(1,2)} }

equipped with the following Lie bracket

[(X_{1}, Z_{1}), (X_{2}, Z_{2})] := ( [X_{1}, X_{2}]_{0}, Z_{2} ^{t}X_{1} − Z_{1} ^{t}X_{2} ),

where [X_{1}, X_{2}]_{0} := X_{1}X_{2} − X_{2}X_{1} denotes the usual matrix bracket and (X_{1}, Z_{1}), (X_{2}, Z_{2}) ∈ 𝔤. And 𝔤 has the following decomposition

𝔤 = 𝔨 ⊕ 𝔭 ( direct sum ),

where

𝔨 := \{ (X, 0) ∈ 𝔤 | X = ( \begin{matrix}0 & x \\ −x & 0 \end{matrix} ) , x ∈ ℝ \}

and

𝔭 := { (X, Z) ∈ 𝔤 | X = ^{t}X ∈ ℝ^{(2,2)}, σ(X) = 0, Z ∈ ℝ^{(1,2)} } .

We observe that 𝔨 is the Lie algebra of K and that we have the following realtions

[𝔨, 𝔨] ⊂ 𝔨  and  [𝔨, 𝔭] ⊂ 𝔭.

Thus the coset space G/K ≅ SP_{2} × ℝ^{(1,2)} is a reductive homogeneous space in the sense of [H2], p. 284. It is easy to see that the adjoint action Ad of K on 𝔭 is given by

(2.6)  Ad (k)((X, Z)) = ( kX^{t}k, Z^{t}k),

where k ∈ K and (X, Z) ∈ 𝔭 with X = ^{t}X, σ(X) = 0. The action (2.6) extends uniquely to the action ρ of K on the polynomial algebra Pol (𝔭)

(2.7)  ρ : K ⟶ Aut ( Pol(𝔭)).

Let Pol (𝔭)^{K} be the subalgebra of Pol (𝔭) consisting of all invariants of the action ρ of K. Then according to [H2], Theorem 4.9, p. 287, there exists a canonical linear bijection λ ( P ⟼ D_{λ(P)}) of Pol (𝔭)^{K} onto 𝔻(SP_{2} × ℝ^{(1,2)}). Indeed, if (ξ_{k}) ( 1 ≤ k ≤ 4 ) is any basis of 𝔭 and P ∈ Pol (𝔭)^{K}, then

(2.8) ( D_{λ(P)}f) (g̃ ∘ (i, 0)) = [ P (\frac{∂}{∂t_{k}} ) f((g̃ ∗ exp (∑_{k=1}^{4} t_{k}ξ_{k})) ∘ (E_{2}, 0))]_{(t_{k})=0} ,

where g̃ ∈ G and f ∈ C^{∞}(SP_{2} × ℝ^{(1,2)}).

We put

e_{1} := ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) , e_{2} := ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) , f_{1} := (1, 0) and f_{2} := (0, 1).

Then e_{1}, e_{2}, f_{1}, f_{2} form a basis of 𝔭. We write for coordinates (X, Z) by

X = ( \begin{matrix}x & y \\ y & −x \end{matrix} ) and Z = (z_{1}, z_{2})

with real variables x, y, z_{1} and z_{2}. Without difficulty we can show that the following polynomials

P (X, Z) : = \frac{1}{8} σ(X^{2}) = \frac{1}{4} (x^{2} + y^{2}),
ξ(X, Z) : = Z^{t}Z = z_{1}^{2} + z_{2}^{2},
P_{1}(X, Z) : = − \frac{1}{2} ZX^{t}Z = \frac{1}{2} (z_{2}^{2} − z_{1}^{2})x − z_{1}z_{2}y

and

P_{2}(X, Z) := \frac{1}{2} (z_{2}^{2} − z_{1}^{2}) y + z_{1}z_{2} x

are algebraically independent generators of Pol (𝔭)^{K}.

Now we are ready to compute the G-invariant differential operators D, Ψ, D_{1} and D_{2} corresponding to the K-invariants P, ξ, P_{1} and P_{2} respectively under the canonical linear bijection (2.8). For real variables t = (t_{1}, t_{2}) and s = (s_{1}, s_{2}), we have

exp ( t_{1}e_{1} + t_{2}e_{2} + s_{1}f_{1} + s_{2}f_{2} )

= ( ( \begin{matrix}a_{1}(t, s) & a_{3}(t, s) \\ a_{3}(t, s) & a_{2}(t, s) \end{matrix} ) , ( b_{1}(t, s), b_{2}(t, s) ) ,

where

a_{1}(t, s) : = 1 + t_{1} + \frac{1}{2!} ( t_{1}^{2} + t_{2}^{2} ) + \frac{1}{3!} ( t_{1}^{3} + t_{1}t_{2}^{2} ) + \frac{1}{4!} (t_{1}^{4} + 2 t_{1}^{2} t_{2}^{2} + t_{2}^{4} ) + ⋯
a_{2}(t, s) : = 1 − t_{1} + \frac{1}{2!} ( t_{1}^{2} + t_{2}^{2} ) − \frac{1}{3!} ( t_{1}^{3} + t_{1}t_{2}^{2} ) + \frac{1}{4!} ( t_{1}^{4} + 2 t_{1}^{2} t_{2}^{2} + t_{2}^{4} ) − ⋯,
a_{3}(t, s) : = t_{2} + \frac{1}{3!} ( t_{1}^{2} t_{2} + t_{2}^{3} ) + \frac{1}{5!} ( t_{1}^{4} t_{2} + 2 t_{1}^{2} t_{2}^{3} + t_{2}^{5} ) + ⋯,
b_{1}(t, s) : = s_{1} − \frac{1}{2!} (s_{1}t_{1} + s_{2}t_{2}) + \frac{1}{3!} s_{1}(t_{1}^{2} + t_{2}^{2}) − \frac{1}{4!} (s_{1}t_{1} + s_{2}t_{2})(t_{1}^{2} + t_{2}^{2}) + ⋯,
b_{2}(t, s) : = s_{2} − \frac{1}{2!} (s_{1}t_{2} − s_{2}t_{1}) + \frac{1}{3!} s_{2}(t_{1}^{2} + t_{2}^{2}) − \frac{1}{4!} (s_{1}t_{2} − s_{2}t_{1})(t_{1}^{2} + t_{2}^{2}) + · · · .

For brevity, we write a_{j}, b_{k} for a_{j}(t, s), b_{k}(t, s) ( j = 1, 2, 3, k = 1, 2 ) respectively. We now fix an element (g, α) ∈ G and write

g = ( \begin{matrix}g_{1} & g_{12} \\ g_{21} & g_{2} \end{matrix} ) ∈ SL(2, ℝ) and α = (α_{1}, α_{2}) ∈ ℝ^{(1,2)}.

We put

( Y (t, s), V (t, s) ) := ( (g, α) ∗ exp ( t_{1}e_{1} + t_{2}e_{2} + s_{1}f_{1} + s_{2}f_{2} )) ∘ (E_{2}, 0)

with

Y (t, s) := ( \begin{matrix}y(t, s)^{−1} & − x(t, s) y(t, s)^{−1} \\ − x(t, s) y(t, s)^{−1} & x(t, s)^{2} y(t, s)^{−1} + y(t, s) \end{matrix} )

and V (t, s) := ( v_{1}(t, s), v_{2}(t, s) ). By an easy calculation, we obtain

x(t, s) = −( ãc̃ + b̃d̃) ( ã^{2} + b̃^{2} )^{−1},
y(t, s) = ( ã^{2} + b̃^{2} )^{−1},
v_{1}(t, s) = ã α̃_{1} + b̃α̃_{2},
v_{2}(t, s) = c̃α̃_{1} + d̃α̃_{2},

where

ã : = g_{1}a_{1} + g_{12}a_{3},
b̃: = g_{1}a_{3} + g_{12}a_{2},
c̃: = g_{21}a_{1} + g_{2}a_{3},
d̃: = g_{21}a_{3} + g_{2}a_{2},
α̃_{1} : = α_{1}a_{2} − α_{2}a_{3} + b_{1},
α̃_{2} : = −α_{1}a_{3} + α_{2}a_{1} + b_{2}.

By an easy calculation, at t = s = 0, we have

\frac{∂x}{∂t_{1}} = 4 g_{1} g_{12} ( g_{1}^{2} + g_{12}^{2} )^{−2},
\frac{∂y}{∂t_{1}} = −2 ( g_{1}^{2} − g_{12}^{2} ) ( g_{1}^{2} + g_{12}^{2} )^{−2},
\frac{∂v_{1}}{∂t_{1}} = \frac{∂v_{2}}{∂t_{1}} = 0,
\frac{∂^{2}x}{∂t_{1}^{2}} = − 16 g_{1} g_{12} ( g_{1}^{2} − g_{12}^{2} ) ( g_{1}^{2} + g_{12}^{2} )^{−3},
\frac{∂^{2}y}{∂t_{1}^{2}} = 8 ( g_{1}^{2} − g_{12}^{2} )^{2} ( g_{1}^{2} + g_{12}^{2} )^{−3} − 4 ( g_{1}^{2} + g_{12}^{2} )^{−1},
\frac{∂^{2}v_{1}}{∂t_{1}^{2}} = \frac{∂^{2}v_{2}}{∂t_{1}^{2}} = 0

and

\frac{∂x}{∂t_{2}} = − 2 ( g_{1}^{2} − g_{12}^{2} ) ( g_{1}^{2} + g_{12}^{2} )^{−2},
\frac{∂y}{∂t_{2}} = − 4 g_{1} g_{12} ( g_{1}^{2} + g_{12}^{2} )^{−2},
\frac{∂v_{1}}{∂t_{2}} = \frac{∂v_{2}}{∂t_{2}} = 0,
\frac{∂^{2}x}{∂t_{2}^{2}} = 16 g_{1} g_{12} ( g_{1}^{2} − g_{12}^{2} ) ( g_{1}^{2} + g_{12}^{2} )^{−3},
\frac{∂^{2}y}{∂t_{2}^{2}} = 32 g_{1}^{2} g_{12}^{2} ( g_{1}^{2} + g_{12}^{2} )^{−3} − 4 ( g_{1}^{2} + g_{12}^{2} )^{−1},
\frac{∂^{2}v_{1}}{∂t_{2}^{2}} = \frac{∂^{2}v_{2}}{∂t_{2}^{2}} = 0.

We note that

ãd̃ − b̃c̃ = 1, a_{1}a_{2} − a_{3}^{2} = 1 and g_{1}g_{2} − g_{12}g_{21} = 1.

We also note that since

( g, α) · (E_{2}, 0) := ( ( \begin{matrix}y^{−1} & 0 \\ 0 & y \end{matrix} ) [( \begin{matrix}1 & −x \\ 0 & 1 \end{matrix} )] , ( v_{1}, v_{2} ) ) ,

we have

y^{−1} = g_{1}^{2} + g_{12}^{2}, x^{2} y^{−1} + y = g_{21}^{2} + g_{2}^{2}, − xy^{−1} = g_{1} g_{21} + g_{12} g_{2}

and

v_{1} = α_{1} g_{1} + α_{2} g_{12}, v_{2} = α_{2} g_{21} + α_{2} g_{2}.

Using the above facts and applying the chain rule, we can easily compute the differential operators D, Ψ, D_{1} and D_{2}. It is known that the images of generators P, ξ, P_{1} and P_{2} under λ are generators of 𝔻(SP_{2} × ℝ^{(1,2)}) ( cf. [H1] or [Y7] ).

Summarizing, we have the following.

Main Theorem. The algebra 𝔻(SP_{2} × ℝ^{(1,2)}) is generated by the following differential operators

(2.9)  D := y^{2} ( \frac{∂^{2}}{∂x^{2}} + \frac{∂^{2}}{∂y^{2}} ) ,

(2.10)  Ψ := y^{−1} \{ \frac{∂^{2}}{∂v_{1}^{2}} − 2 x \frac{∂^{2}}{∂v_{1}∂v_{2}} + ( x^{2} + y^{2} ) \frac{∂^{2}}{∂v_{2}^{2}} \} ,

(2.11)  D_{1} := \{ \frac{∂^{2}}{∂v_{1}^{2}} − 2x \frac{∂^{2}}{∂v_{1}∂v_{2}} + (x^{2} − y^{2}) \frac{∂^{2}}{∂v_{2}^{2}} \} \frac{∂}{∂y}

+ 2y ( \frac{∂^{2}}{∂v_{1}∂v_{2}} − x \frac{∂^{2}}{∂v_{2}^{2}} ) \frac{∂}{∂x} − Ψ

and

(2.12)  D_{2} := \{ \frac{∂^{2}}{∂v_{1}^{2}} − 2x \frac{∂^{2}}{∂v_{1}∂v_{2}} + (x^{2} − y^{2}) \frac{∂^{2}}{∂v_{2}^{2}} \} \frac{∂}{∂x}

− 2y ( \frac{∂^{2}}{∂v_{1}∂v_{2}} − x \frac{∂^{2}}{∂v_{2}^{2}} ) \frac{∂}{∂y} ,

where Y is of the form (2.4) and V = ( v_{1}, v_{2} ) with real variables x, y, v_{1}, v_{2} and y > 0. Moreover, we have

[D, Ψ] : = D Ψ − Ψ D

= − 4 y \frac{∂^{3}}{∂x∂v_{1}∂v_{2}} + 4 x y \frac{∂^{3}}{∂x∂v_{2}^{2}} − 2 \frac{∂^{3}}{∂y∂v_{1}^{2}} + 4 x \frac{∂^{3}}{∂y∂v_{1}∂v_{2}}

+ 2 ( y^{2} − x^{2} ) \frac{∂^{3}}{∂y∂v_{2}^{2}} + \frac{2 x}{y} \frac{∂^{2}}{∂v_{1}∂v_{2}} + 2 Ψ.

In particular, the algebra 𝔻(SP_{2} × ℝ^{(1,2)}) is not commutative.

Now we present a G-invariant metric on SP_{2} × ℝ^{(1,2)}.

Theorem 2.2. The following Riemannian metric ds^{2} on SP_{2} × ℝ^{(1,2)} defined by

ds^{2} := \frac{1}{y^{2}} ( dx^{2} + dy^{2} ) + \frac{1}{y} \{ ( x^{2} + y^{2} ) dv_{1}^{2} + 2 x dv_{1} dv_{2} + dv_{2}^{2} \}

is invariant under the action (1.2) of G, where we used the coordinates ( Y, V ) such that Y is of the form (2.4) and V = ( v_{1}, v_{2} ) with v_{1}, v_{2} real variables. The Laplace-Beltrami operator Δ of the Riemannian space ( SP_{2} × ℝ^{(1,2)}, ds^{2} ) is given by

Δ = y^{2} ( \frac{∂^{2}}{∂x^{2}} + \frac{∂^{2}}{∂y^{2}} ) + \frac{1}{y} \{ \frac{∂^{2}}{∂v_{1}^{2}} − 2 x \frac{∂^{2}}{∂v_{1}∂v_{2}} + ( x^{2} + y^{2} ) \frac{∂^{2}}{∂v_{2}^{2}} \} .

That is, Δ = D + Ψ. In particular, G is a subgroup of the isometry group of ( SP_{2} × ℝ^{(1,2)}, ds^{2} ).

INVARIANT DIFFERENTIAL OPERATORS 9

Proof. We put

dY := ( \begin{matrix}− y^{−2} dy & − y^{−1} dx + x y^{−2} dy \\ − y^{−1} dx + x y^{−2} dy & 2 x y^{−1} dx + ( 1 − x^{2} y^{−2} ) dy \end{matrix} ) .

Then since

Y^{−1} = ( \begin{matrix}y + x^{2} y^{−1} & x y^{−1} \\ x y^{−1} & y^{−1} \end{matrix} ) ,

we can easily show that

ds^{2} = \frac{1}{2} σ(Y^{−1}dY Y^{−1}dY) + dV Y^{−1} ^{t}(dV).

For an element ( g, α) ∈ G with g ∈ SL(2, ℝ) and α ∈ ℝ^{(1,2)}, we put

( Y^{∗}, V^{∗}) := ( g, α) · (Y, V) = ( gY ^{t}g, (V + α)^{t}g).

Since

Y^{∗} = gY ^{t}g and V^{∗} = ( V + α)^{t}g,

we get

dY^{∗} = g dY ^{t}g and dV^{∗} = ( V + α)^{t}g.

Therefore by a simple calculation, we can show that σ ( Y^{∗−1} dY^{∗} Y^{∗−1} dY^{∗} ) + dV^{∗} Y^{∗−1} ^{t}(dV^{∗})

= σ( Y^{−1}dY Y^{−1} dY ) + dV Y^{−1} ^{t}(dV).

Hence the metric ds^{2} is invariant under the action (1.2) of G.

Since the matrix form ( g_{ij} ) of the metric ds^{2} is given by

( g_{ij} ) = ( \begin{matrix}y^{−2} & 0 & 0 & 0 \\ 0 & y^{−2} & 0 & 0 \\ 0 & 0 & y^{−1} ( x^{2} + y^{2} ) & x y^{−1} \\ 0 & 0 & x y^{−1} & y^{−1} \end{matrix} ) ,

the inverse matrix ( g^{ij} ) of ( g_{ij} ) is given by

( g^{ij} ) = ( \begin{matrix}y^{2} & 0 & 0 & 0 \\ 0 & y^{2} & 0 & 0 \\ 0 & 0 & y^{−1} & − x y^{−1} \\ 0 & 0 & − x y^{−1} & (x^{2} + y^{2}) y^{−1} \end{matrix} ) .

Hence by an easy calculation, we can show that the Laplace-Beltrami operator Δ is equal to D + Ψ. □

10 JAE-HYUN YANG

3. The Lie derivatives on G

In this section, we compute the Lie derivatives for functions on G and SP_{2}×ℝ^{(1,2)} explicitly.

We put

W_{1} := ( ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) , (0, 0) ) , W_{2} := ( ( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ) , (0, 0) ) , W_{3} := ( ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) , (0, 0))

and

W_{4} := ( ( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} ) , (1, 0) ) , W_{5} := ( ( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} ) , (0, 1) ) .

Clearly W_{1}, ⋯, W_{5} form a basis of 𝔤.

We define the differential operators L_{k}, R_{k} ( 1 ≤ k ≤ 5 ) on G by

(3.1)  L_{k}f(g̃) := \frac{d}{dt}|_{t=0}f(g̃ ∗ exp tW_{k})

and

(3.2)  R_{k}f(g̃) := \frac{d}{dt}|_{t=0}f(exp tW_{k} ∗ g̃),

where f ∈ C^{∞}(G) and g̃ ∈ G.

We also put

H := ( ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) , (0, 0) ) and V := ( ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) , (0, 0) ) .

We define the differential operators L_{H}, L_{V} , R_{H} and R_{V} by

L_{H}f(g̃) : = \frac{d}{dt}|_{t=0}f(g̃ ∗ exp tH),
L_{V} f(g̃) : = \frac{d}{dt}|_{t=0}f(g̃ ∗ exp tV),
R_{H}f(g̃) : = \frac{d}{dt}|_{t=0}f(exp tH ∗ g̃)

and

R_{V} f(g̃) := \frac{d}{dt}|_{t=0}f(exp tV ∗ g̃).

By an easy calculation, we get

exp tW_{1} = ( ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) , (0, 0) ) , exp tW_{2} = ( ( \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ) , ( 0, 0 ) )
exp tW_{3} = ( ( \begin{matrix}e^{t} & 0 \\ 0 & e^{−t} \end{matrix} ) , ( 0, 0 ) ) , exp tW_{4} = ( ( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} ) , ( t, 0 ) )

INVARIANT DIFFERENTIAL OPERATORS 11

and

exp tW_{5} = ( ( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} ) , ( 0, t) ) .

Now we use the following coordinates (g, α) in G given by

(3.3)  g = ( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) ( \begin{matrix}y^{1/2} & 0 \\ 0 & y^{−1/2} \end{matrix} ) ( \begin{matrix}cos θ & sin θ \\ −sin θ & cos θ \end{matrix} )

and

(3.4)  α = ( α_{1}, α_{2} ),

where x, α_{1}, α_{2} ∈ ℝ, y > 0 and 0 ≤ θ < 2π. By an easy computation, we have

L_{1} = y cos 2θ \frac{∂}{∂x} + y sin 2θ \frac{∂}{∂y} + sin^{2} θ \frac{∂}{∂θ} − α_{2} \frac{∂}{∂α_{1}} ,
L_{2} = y cos 2θ \frac{∂}{∂x} + y sin 2θ \frac{∂}{∂y} − cos^{2} θ \frac{∂}{∂θ} − α_{1} \frac{∂}{∂α_{2}} ,
L_{3} = − 2y sin 2θ \frac{∂}{∂x} + 2 y cos 2θ \frac{∂}{∂y} + sin 2θ \frac{∂}{∂θ} − α_{1} \frac{∂}{∂α_{1}} + α_{2} \frac{∂}{∂α_{2}} ,
L_{4} = \frac{∂}{∂α_{1}} ,
L_{5} = \frac{∂}{∂α_{2}} ,
R_{1} = \frac{∂}{∂x},
R_{2} = ( y^{2} − x^{2} ) \frac{∂}{∂x} − 2 xy \frac{∂}{∂y} − y \frac{∂}{∂θ} ,
R_{3} = 2 x \frac{∂}{∂x} + 2 y \frac{∂}{∂y} ,
R_{4} = y^{−1/2} cos θ \frac{∂}{∂α_{1}} + y^{−1/2} sin θ \frac{∂}{∂α_{2}} ,
R_{5} = − y^{−1/2} ( x cos θ + y sin θ) \frac{∂}{∂α_{1}} + y^{−1/2} ( y cos θ − x sin θ) \frac{∂}{∂α_{2}} .

Since

H = W_{1} − W_{2} and V = W_{1} + W_{2},

we have

L_{H} = L_{1} − L_{2} = \frac{∂}{∂θ} − α_{2} \frac{∂}{∂α_{1}} + α_{1} \frac{∂}{∂α_{2}} ,
L_{V} = L_{1} + L_{2}

= 2 y cos 2θ \frac{∂}{∂x} + 2 y sin 2θ \frac{∂}{∂y} − cos 2θ \frac{∂}{∂θ} − α_{2} \frac{∂}{∂α_{1}} − α_{1} \frac{∂}{∂α_{2}} ,

R_{H} = R_{1} − R_{2}

= ( 1 − y^{2} + x^{2}) \frac{∂}{∂x} + 2 xy \frac{∂}{∂y} + y \frac{∂}{∂θ}

and

R_{V} = R_{1} + R_{2}

= ( 1 + y^{2} − x^{2} ) \frac{∂}{∂x} − 2 xy \frac{∂}{∂y} − y \frac{∂}{∂θ} .

For instance, we present a calculation for L_{3} and R_{5}. For a simplicity, we denote the form (3.3) by [x, y, θ]. Let (g, α) be an element of SL_{2,1}(ℝ) with

g := ( \begin{matrix}a & b \\ c & d \end{matrix} ) = [ x, y, θ] and α := (α_{1}, α_{2}) ∈ ℝ^{(1,2)}. Then by an easy computa-

tion, we have

(3.5)  d − i c = y^{−1/2} e^{iθ} and d + i c = y^{−1/2} e^{−iθ}.

So we get

| d + i c | = y^{−1/2} and e^{iθ} = \frac{d − i c}{| d + i c |}.

We put

( g(t), α(t) ) := [ x_{∗}, (t), y_{∗}(t), θ_{∗}(t) ] := (g, α) ∗ exp tW_{3}, t ∈ ℝ

and

τ_{∗}(t) := g(t) < i > = x_{∗}(t) + i y_{∗}(t), t ∈ ℝ.

Then

g(t) = ( \begin{matrix}a e^{t} & b e^{−t} \\ c e^{t} & d e^{−t} \end{matrix} ) and α(t) = ( α_{1}e^{−t}, α_{2} e^{t}).

Thus

τ_{∗}(t) = \frac{i a e^{t} + b e^{−t}}{i c e^{t} + d e^{−t}} = \frac{i a e^{2t} + b}{i c e^{2t} + d} = x_{∗}(t) + i y_{∗}(t).

Differentiating τ_{∗}(t) with respect to t and evaluating at t = 0, by (3.5), we have

τ′_{∗}(0) = 2 i y e^{2iθ}.

Thus

x′_{∗}(0) = − 2 y sin 2θ and y′_{∗}(0) = 2 y cos 2θ.

And since

i θ_{∗}(t) = log \frac{d e^{−t} − i c e^{t}}{( c^{2} e^{2t} + d^{2} e^{−2t})^{1/2}} ,

we obtain

θ′_{∗}(0) = 2 sin θ · cos θ = sin 2θ.

We set

α(t) := ( α_{1,∗}(t), α_{2,∗}(t) ), t ∈ ℝ.

Since

α_{1,∗}(t) = α_{1} e^{−t} and α_{2,∗}(t) = α_{2} e^{t},

we get

α′_{1,∗}(0) = − α_{1} and α′_{2,∗}(0) = α_{2}.

For a smooth function f on SL_{2,1}(ℝ), using the chain rule, we get the desired formula for L_{3}.

INVARIANT DIFFERENTIAL OPERATORS 13

Since

exp tW_{5} = ( E_{2}, (0, t) ),

we have

exp tW_{5} ∗ (g, α) = ( g, (0, t)^{t}g^{−1} + α).

We put

( g(t), α(t) ) = exp tW_{5} ∗ (g, α)

and

g(t) = [ x_{∗}(t), y_{∗}(t), θ_{∗}(t) ], α(t) = ( α_{1,∗}(t), α_{2,∗}(t) ).

Then we have

g(t) = g, α_{1,∗}(t) = −b t + α_{1} and α_{2,∗}(t) = a t + α_{2}.

From (3.3), we get

a = y^{1/2} cos θ − x y^{−1/2} sin θ and b = y^{1/2} sin θ + x y^{−1/2} cos θ.

Finally we obtain the desired explicit formula for R_{5}. We leave the other formulas to the reader.

We define the differential operators 𝕃_{k} ( 1 ≤ k ≤ 5 ) on SP_{2} × ℝ^{(1,2)} by

𝕃_{k}f(Y, V) := \frac{d}{dt}|_{t=0} f(exp tW_{k} · (Y, V)), 1 ≤ k ≤ 5,

where f ∈ C^{∞}(SP_{2} × ℝ^{(1,2)}) and (Y, V) ∈ SP_{2} × ℝ^{(1,2)}. Proposition 3.1. The differential operators 𝕃_{k}’s are given by

𝕃_{1} = ( x^{2} − y^{2} ) \frac{∂}{∂x} + 2 x y \frac{∂}{∂y} + v_{2} \frac{∂}{∂v_{1}}
𝕃_{2} = − \frac{∂}{∂x} + v_{1} \frac{∂}{∂v_{2}} ,
𝕃_{3} = − 2 x \frac{∂}{∂x} − 2 y \frac{∂}{∂y} + v_{1} \frac{∂}{∂v_{1}} − v_{2} \frac{∂}{∂v_{2}} ,
𝕃_{4} = \frac{∂}{∂v_{1}} ,
𝕃_{5} = \frac{∂}{∂v_{2}} .

Proof. It is easy to prove it and so we omit its proof. □

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Max-Planck-Institut für Mathematik Gottfried-Claren-Strasse 26 53225 Bonn Germany

email : yang@mpim-bonn.mpg.de